4. 3D Solid Mechanics#
The equations of elasticity describe the deformation of solids due to applied forces. The treatment by the finite element method is very similar.
from netgen.occ import *
from ngsolve import *
from ngsolve.webgui import Draw
box = Box((0,0,0), (3,0.6,1))
box.faces.name="outer"
cyl = sum( [Cylinder((0.5+i,0,0.5), Y, 0.25,0.8) for i in range(3)] )
cyl.faces.name="cyl"
geo = box-cyl
Draw(geo);
find edges between box and cylinder, and build chamfers:
cylboxedges = geo.faces["outer"].edges * geo.faces["cyl"].edges
cylboxedges.name = "cylbox"
geo = geo.MakeChamfer(cylboxedges, 0.03)
name faces for boundary conditions:
geo.faces.Min(X).name = "fix"
geo.faces.Max(X).name = "force"
Draw(geo);
mesh = Mesh(OCCGeometry(geo).GenerateMesh(maxh=0.1)).Curve(3)
Draw (mesh);
4.1. Linear elasticity#
Displacement: \(u : \Omega \rightarrow R^3\)
Linear strain: $\( \varepsilon(u) := \tfrac{1}{2} ( \nabla u + (\nabla u)^T ) \)$
Stress by Hooke’s law: $\( \sigma = 2 \mu \varepsilon + \lambda \operatorname{tr} \varepsilon I \)$
Equilibrium of forces: $\( \operatorname{div} \sigma = f \)$
Displacement boundary conditions: $\( u = u_D \qquad \text{on} \, \Gamma_D \)$
Traction boundary conditions: $\( \sigma n = g \qquad \text{on} \, \Gamma_N \)$
4.2. Variational formulation:#
Find: \(u \in H^1(\Omega)^3\) such that \(u = u_D\) on \(\Gamma_D\) $\( \int_\Omega \sigma(\varepsilon(u)) : \varepsilon(v) \, dx = \int_\Omega f v dx + \int_{\Gamma_N} g v ds \)\( holds for all \)v = 0\( on \)\Gamma_D$.
E, nu = 210, 0.2
mu = E / 2 / (1+nu)
lam = E * nu / ((1+nu)*(1-2*nu))
def Stress(strain):
return 2*mu*strain + lam*Trace(strain)*Id(3)
fes = VectorH1(mesh, order=3, dirichlet="fix")
u,v = fes.TnT()
gfu = GridFunction(fes)
with TaskManager():
a = BilinearForm(InnerProduct(Stress(Sym(Grad(u))), Sym(Grad(v))).Compile()*dx)
pre = Preconditioner(a, "bddc")
a.Assemble()
force = CF( (1e-3,0,0) )
f = LinearForm(force*v*ds("force")).Assemble()
from ngsolve.krylovspace import CGSolver
inv = CGSolver(a.mat, pre, tol=1e-8)
gfu.vec.data = inv * f.vec
with TaskManager():
fesstress = MatrixValued(H1(mesh,order=3), symmetric=True)
gfstress = GridFunction(fesstress)
gfstress.Interpolate (Stress(Sym(Grad(gfu))))
Look at deformed body, use slider deformation in gui controls:
Draw (5e4*gfu, mesh);
The stresses show maximal internal material load:
Draw (Norm(gfstress), mesh, deformation=1e4*gfu, draw_vol=False, order=3);
Exercise:
apply loading in \(z\)-direction.
look at individual components of stress-tensor (like
gfstress[0,0])